Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

Let $S$ the set of natural numbers from $1$ up to $1001$ , $S=\{1,2,...,1001\}$. Lisandro thinks of a number $N$ of $S$ , and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of $S$, Lisandro reads it and tells Carla how many subsets of her list contain $N$ . If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than $3$ are allowed. What is the smallest total number of subsets that allow Carla to find $N$ for sure?

Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$. I. Bogdanov

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician. Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?

Tao plays the following game:given a constant $v>1$;for any positive integer $m$,the time between the $m^{th}$ round and the $(m+1)^{th}$ round of the game is $2^{-m}$ seconds;Tao chooses a circular safe area whose radius is $2^{-m+1}$ (with the border,and the choosing time won't be calculated) on the plane in the $m^{th}$ round;the chosen circular safe area in each round will keep its center fixed,and its radius will decrease at the speed $v$ in the rest of the time(if the radius decreases to $0$,erase the circular safe area);if it's possible to choose a circular safe area inside the union of the rest safe areas sometime before the $100^{th}$ round(including the $100^{th}$ round),then Tao wins the game.If Tao has a winning strategy,find the minimum value of $\biggl\lfloor\frac{1}{v-1}\biggr\rfloor$.

Given a (simple) graph $G$ with $n \geq 2$ vertices $v_1, v_2, \dots, v_n$ and $m \geq 1$ edges, Joël and Robert play the following game with $m$ coins: [list=i] [*]Joël first assigns to each vertex $v_i$ a non-negative integer $w_i$ such that $w_1+\cdots+w_n=m$. [*]Robert then chooses a (possibly empty) subset of edges, and for each edge chosen he places a coin on exactly one of its two endpoints, and then removes that edge from the graph. When he is done, the amount of coins on each vertex $v_i$ should not be greater than $w_i$. [*]Joël then does the same for all the remaining edges. [*]Joël wins if the number of coins on each vertex $v_i$ is equal to $w_i$. [/list] Determine all graphs $G$ for which Joël has a winning strategy.

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?